An iterative contractive framework for probe methods: LASSO

Authors


Abstract

[1] We present a new iterative approach called Line Adaptation for the Singular Sources Objective (LASSO) to object or shape reconstruction based on the singular sources method (or probe method) for the reconstruction of scatterers from the far-field pattern of scattered acoustic or electromagnetic waves. The scheme is based on the construction of an indicator function given by the scattered field for incident point sources in its source point from the given far-field patterns for plane waves. The indicator function is then used to drive the contraction of a surface which surrounds the unknown scatterers. A stopping criterion for those parts of the surfaces that touch the unknown scatterers is formulated. A splitting approach for the contracting surfaces is formulated, such that scatterers consisting of several separate components can be reconstructed. Convergence of the scheme is shown, and its feasibility is demonstrated using a numerical study with several examples.

1. Introduction

[2] Inverse scattering theory has a long history with classical contributions for example by Lax and Phillips [1967]. An introduction into the theory of acoustic and electromagnetic inverse scattering can be found in the work of Colton and Kress [1998] or Kirsch [1996]. More recently, new classes of methods have been introduced with sampling and probe methods [see Cakoni and Colton, 2006; Kirsch and Grinberg, 2008; Potthast, 2006, 2001]. Powerful sampling schemes have been introduced by Colton and Kirsch [1996] with the linear sampling method and by Kirsch [1998] with the factorization method. Further highly interesting methods are the probe method [Ikehata, 1998], the singular sources method [Potthast, 2001], the no response test [Luke and Potthast, 2003], the range test [Potthast et al., 2003] and Ikehata's enclosure method, compare with the survey by Potthast [2006]. Each of these methods exploits different properties of the scattering map or particular scattered fields which are then reconstructed from the measurements.

[3] The main idea of many methods is to formulate an indicator function μ defined either in the space equation imagem or on a set of test domains. This function characterizes the unknown scatterers, their physical properties or their shape.

[4] Here, we will study a new realization of the singular sources method first introduced by Potthast [2000]. It employs the modulus μ(z) ≔ ∣Φs(z, z)∣ of the scattered field Φs(z, z) for source points in some point z with evaluation of the scattered field in z. The basic background of the singular sources method will be introduced in detail in section 2. The main question we need to address arises from the way in which the indicator function μ(z) for some zequation image2 is calculated for probe methods. Usually, some approximation domain G is used, where zequation image is needed. Convergence for the reconstruction of μ(z) can be shown if the unknown inclusion D is a subset of the approximation domain G. If D is unknown, clearly the key algorithmical problem is the choice of approximation domains G which are adapted to the knowledge about the unknown scatterers throughout the reconstruction procedure.

[5] We will study three different methods, two of which have been proposed in the literature [Ikehata, 1998; Potthast and Schulz, 2007]. We will discuss advantages and disadvantages of the needle approach and of domain sampling and then, with the Line Adaptation for the Singular Sources Objective (LASSO) scheme, study a novel iterative approach. The basic idea of the LASSO scheme is to construct a contractive curve or surface, respectively, which always contains the unknown scatterers in its interior.

[6] In section 2 we survey the singular sources method on which the curve iterations are based and provide all necessary definitions. Section 3.3 serves to provide an introduction into the algorithmical setup which can be employed for the singular sources method or more general probe methods. Numerical examples will be presented in section 4 which prove the feasibility of the method. Here, for simplicity we restrict our attention to the two-dimensional case.

2. Singular Sources Method

[7] We study time-harmonic scattering of acoustic or electromagnetic waves in two dimensions (see Colton and Kress [1998] for a detailed introduction). Here, our task is the determination of some (sound soft or sound hard) scatterer D from the knowledge of the far-field pattern u(equation image, d), equation image, dequation imageequation image for all incident plane waves ui(·, d), dequation imageequation image. We will employ the notation us(y, d), yequation image2 \ D, dequation image for the scattered field of the incident plane waves ui(·, d). Here, we will restrict our attention to the two-dimensional case and to scattered fields us satisfying the Helmholtz equation

equation image

and the Sommerfeld radiation condition

equation image

[8] In the simplest case the singular sources method is based on scattering by a point source

equation image

with source point zequation imageequation image2 \ equation image. The scattered field for this singular source is denoted as Φs(x, z) for xequation imageequation image2. For scattering by a sound-soft obstacle from the boundary condition Φs(x, z) = −Φ(x, z) for x ∈ ∂D, zequation image2 \ equation image we have

equation image

The convergence analysis for the singular sources method as worked out by Potthast [2001, 2006] shows that we have

equation image

[9] The idea of the singular sources method is to use the behaviour (5) to detect the unknown shape ∂D. The indicator function

equation image

for z in subsets of equation image2 \ equation image can be approximately constructed from the far-field pattern or measured scattered field. The reconstruction of (6) from u is carried out in several steps.

[10] 1. We first choose some approximation domain G with equation imageG and zequation image. Then we use the point source approximation

equation image

with some density gzequation imageL2 (equation image), i.e. we construct a kernel gz for a Herglotz wave function (as defined by Colton and Kress [1998]) to approximate the point source on the domain of approximation G. The approximation in (7) is to be understood in the sense that for every ε > 0 we can find a density gzL2 (equation image) such that

equation image

Practically, the density gz is determined by solving a boundary integral equation

equation image

which is known as point source equation and is the key equation employed for the point source method [Potthast, 2001, 1996, 1998]. We employ the abbreviation

equation image

for the Herglotz wave function considered as an incident field. We need to remark that equation (9) cannot not have a solution, but denseness results as proven by Potthast [2001] show that approximate solutions are possible and can be found using Tikhonov regularization.

[11] 2. As the second step we note that from the approximation (7) or (8), respectively, we derive a corresponding approximation for the scattered fields and for the far-field pattern for the incident fields Φ (·, z) and wi. First, for the scattered fields obtain the approximation

equation image

with the density gzequation imageL2 (equation image) with the density from above, which is varied due to the well-posedness of the scattering problem. Second, passing on both sides of (11) to the corresponding the far-field patterns we derive

equation image

Here, the approximations ≈ are understood in the sense that given ε > 0 there is a density gz such that the difference between the right-hand side of the equation and its left-hand side is smaller than ε. The field ws[gz] is the scattered field for the incident field wi[gz] defined in (10). Its far-field pattern is given by w[gz].

[12] 3. Finally, we apply the Point Source Method (as introduced to reconstruct (11) above) to the far-field patterns Φ(·, z) and w given in (12) to reconstruct the scattered field Φs(x, z) for xequation imageequation image2 \ equation image and the field ws on xequation imageequation image2 \ equation image (see Potthast [2001] for more details). Then we obtain

equation image

for x, zequation image2 \ equation image in the sense defined above. For ε → 0 in (8) the right-hand side of (13) converges to the left-hand side [cf. Potthast, 2001].

[13] We show the indicator function for the singular sources method for a kite shaped domain in Figure 1 and for two or four domains, respectively, it is shown in Figure 2.

Figure 1.

The indicator function for the singular sources method for a kite shaped domain. The reconstruction has been carried out using domain sampling.

Figure 2.

We show a simulation of the indicator function (6) used by the singular sources or probe method, respectively, for (a) two kite shaped objects and (b) four objects with different shapes. The objects are clearly visible in dark red.

3. Algorithmical Setup for Probe Methods

[14] The reconstruction of the indicator function (6) of the singular sources method (as for the probe method) needs to employ approximation domains G for which the unknown scatterer D is inside of G, i.e.

equation image

Though (14) is a sufficient condition only, if it is violated in general we do not get convergence of the reconstruction for the indicator function following (13). To guarantee this condition and then employ the reconstruction we can follow different approaches. For our following arguments we will assume D to be a subset of some ball BR with radius R > 0.

3.1. The Needle Approach

[15] The idea of the needle approach is to use curves (needles) which probe the area under consideration. In the simplest case this can be a straight line from outside of BR to some point zBR. If we let G be the complement of this line in BR, then if we start with z far away from the scatterer a, then as long as this line is outside of the scatterer D and does not touch it, we obtain convergence of the reconstruction of μ(z) in z.

[16] There are advantages of the needle approach: The needle approach is a nice theoretical tool to study the convergence of the method. Needles are thin manifolds and can reach the area between different scatterers or scatterers which consist of different separate components. However, numerically, the needle approach is highly unstable due to the strong non-convexity of the corresponding approximation domains G. Also, since the domains G are strongly changing throughout the probing procedure, the numerical approximation of the indicator functions is quite unstable, leading to rather bad practical reconstruction results.

3.2. Domain Sampling

[17] As an alternative to the needle approach, domain sampling has been introduced by Potthast et al. [2003] and Potthast [2004]. The idea of domain sampling is to carry out a reconstruction for many different domains G and test the convergence of the reconstructions on the boundary ∂G of these domains or on sets outside of G. If the reconstruction is convergent in some area, we use the corresponding reconstruction. We then combine all the good reconstructions by masking operations and taking pointwise weighted averages.

[18] Domain sampling is a very stable algorithm for which convergence can be shown [Potthast and Schulz, 2007]. If the set of sampling domains is chosen sufficiently rich, domain sampling leads to very good reconstructions as shown in Figure 1. However, the disadvantage of this approach is that the reconstructions need to be carried out for many domains. This needs time and the algorithm looses much of its potentially high efficiency.

3.3. LASSO Scheme

[19] An alternative to domain sampling keeping its stability is the following LASSO approach. We define an iterative procedure which is driven by the probe functional. However, now the indicator function is reconstructed only on the boundary of the iteration domains Gn for n = 1, 2, 3, … We start with some large domain, for example with G0BR. Then, we move the boundary of G inward for all points x ∈ ∂G where the indicator function μ(x) is smaller than some threshold. This guarantees that (14) is satisfied.

[20] The above mentioned contractions are combined with smoothing of the remaining curve in each step and with splitting if two parts of the contracting curve touch each other. We next describe details of the algorithm, where the splitting procedure can be found in section 3.4. A numerical study for the LASSO scheme is carried out in section 4.

[21] Next, we describe one step of the iterative LASSO algorithms. Assume that Gn in the nth step of the algorithm, nequation imageequation image, is an approximation domain which satisfies (14). The goal is to obtain a new approximation domain Gn+1 such that (14) is satisfied and such that we have the contraction property

equation image

Here we assume that Gn+1Gn; that is, we have a strict inclusion, though we do not necessarily assume that equation imageGn; that is, some parts of the boundary ∂Gn+1 of Gn+1 might coincide with parts of the boundary ∂Gn of Gn. If in fact Gn+1 = Gn in some approximate sense, we will stop our iterations; that is, the stopping criterion we employ is

equation image

with some appropriate distance equation image > 0, where d(·,·) denotes the Haussdorf distance

equation image

The construction of Gn+1 is carried out as follows. In the first step we define a new intermediate domain by a contraction of Gn. We will assume that Gn has a boundary of class C1 and denote the exterior unit normal by ν. Then, given some step size h > 0 the domain equation imagen+1 is defined by its boundary

equation image

For h sufficiently small, the domain equation imagen+1 satisfies equation imagen+1Gn and its boundary is of class C0.

[22] We need to guarantee that DGn+1. We know we have ∣Φs(z, z)∣ → ∞ when z → ∂D. As long as Dequation imagen+1 we know that we can reconstruct Φs(z, z) on ∂equation imagen+1. Further, we know that when a point zequation imageequation imagen+1,h approaches ∂D, then the reconstruction Φαs(z, z) of Φs(z, z) must become large in that point. We use this condition now to define the following indicator function

equation image

The constant c is one of the basic parameters of the singular sources method. It is usually chosen to be adequate for some circular reference domain to provide a good reconstruction for a given regularization method when reconstructing Φs(z, z) and is then kept fixed. We use this to define

equation image

The set equation imagen+1 consists of parts which are subsets of ∂Gn and parts which are subsets of ∂equation imagen+1, depending on whether the indicator function (19) shows that we reached the boundary of the unknown domain D already.

[23] The set equation imagen+1 is piecewise smooth, but it has jumps of size h where the indicator function η jumps. To achieve a smooth domain we now apply a smoothing algorithms, providing a smooth approximation Λn+1 = ∂Gn+1 to equation imagen. The domain Gn+1 is then next iterate for the LASSO scheme as long as we do not need to apply the splitting described in section 3.4.

[24] There are several methods which can be used for smoothing of a m−1 dimensional manifold equation imageequation image2 for mequation imageequation image. We have employed the convolution type mapping

equation image

with a constant

equation image

and Γ: = {S(x) : xequation imageequation image}, where we used Gaussian kernels

equation image

with decay rate τ > 0 for ℓ = 0 or ℓ = 1. Stronger smoothing can be achieved by iterating equation image, i.e., taking ��L with some Lequation image. As an alternative smoothing we can employ a smooth approximation to η in (19), which will lead to a smooth update of the approximation domain.

[25] For a discretization of a curve in two dimensions, as used for the numerical examples in section 4, an approximate version of the smoothing (21) for ℓ = 0 is calculated as follows. Let p = (p1, p2, …, pn) be an ordered set of n points pjequation imageequation image2 representing a curve Λn+1, for example by using some global parametrization ψ over [0, 2π) and pj = ψ(2π/n(j − 1)). For an appropriate choice of k the discretized version of the above integrals leads to smoothing formulas such as

equation image

where mod(j, n) maps j into the set {1, ., n} modulus n. In the case ℓ = 1 we obtain formulas similar to

equation image

We usually repeat the schemes (24) or (25), respectively, several times. The number of repetitions controls the degree of smoothing which is achieved, it is used as a smoothing parameter.

3.4. Domain Splitting

[26] The pure contraction as described in section 3.3 does not allow changes in the topology of the curve ∂D. A solution to this is domain splitting where we change the topology of the curve when ever it touches itself in some point x0 ∈ ∂Gn. To identify a point where a curve ∂Gn touches itself, we need to setup a proper distance of a curve to itself.

[27] 1. We use some periodic parametrization γ : [0, 2π) → equation image2, tγ(t) for ∂Gn with period 2π. Then for t ∈ [0, 2π) we define

equation image

which is the surface which does not include a neighborhood of the point γ(t) when neighborship is defined via the manifold [0, 2π) modulus 2π. The parameter ρ is has been chosen by trial and error so far, it is usually some fraction like 1/3 or 1/5 of a typical diameter of one of the components of the scatterer under consideration.

[28] 2. When ever a point z = γ(t) comes close to Γρ(t), we conclude that the curve is touching itself. Then, we split the curve at γ(t) and remove all points which are in a neighborhood of γ(t). We usually chose the ball of radius 2ρ here.

[29] 3. The remaining points are collected into two new curves Γ1 and Γ2. These curves are smoothed separately, such that we have a new curve which consists of two separate smooth components.

[30] We start with one curve Γ. After the first splitting it consists of two curves Γ1 and Γ2. If for example Γ1 is split again, we obtain Γ1,1, Γ1,2 and Γ2. A demonstration of the splitting step can be seen in Figures 5 and 6.

4. Numerical Examples

[31] The LASSO scheme starts with some large curve and then defines contractions step by step controlled by a reconstruction of the singular sources or probe method, respectively. Here, we show examples which demonstrate the feasibility and power of the tools described in sections 3.3 and 3.4.

[32] The following numerical examples have been carried out using an integral equation approach for the forward problem as described by Kress [1989]. With the far-field data u(equation image, d) for equation image, dequation image we then carried out the LASSO reconstruction steps. For every curve which is now given by a vector of discrete points zk, k = 1, ., N we carried out the reconstruction of μ(zk) at a point zz + (z) with a small fixed parameter r > 0. For our numerics we have usually chosen r = 0.1, where our scatterers have diameter of approximately 1. We have experimented with the step size h, chosen as h = 0.1, h = 0.2 and h = 0.5. The size of h will determine the precision of the reconstruction, since the iteration tests points of distance h and then stops if the stopping criterion is satisfied. Clearly, a small h needs many steps until the curves reach the scatterers. This suggests an adaptive use of h, which is beyond the scope of this work.

[33] We first demonstrate the indicator function in Figure 2. The unknown objects can be clearly seen in dark red. Here, we need to keep in mind that a direct reconstruction of the indicator function is not possible, but that a sufficient condition is that the unknown object is inside of some approximation domain and a reconstruction is possible outside of this approximation domain. Figures 3–6 show some selections of the steps from the full reconstruction process.

Figure 3.

Demonstration of iterative curve defined by the LASSO scheme for the reconstruction of two separate objects in steps 10 and 20. The unknown obstacles are indicated in light gray.

Figure 4.

Demonstration of iterative curve defined by the LASSO scheme for the reconstruction of two separate objects in steps 40 and 50. The unknown obstacles are indicated in light gray. Here the shape of the two objects starts to be visible.

Figure 5.

Demonstration of iterative curve defined by the LASSO scheme for the reconstruction of two separate objects in steps 56, 57 and 58. We demonstrate the steps where the curve touches itself and where splitting is applied.

Figure 6.

Demonstration of iterative curve defined by the LASSO scheme for the reconstruction of two separate objects in steps 59 and 66. Smoothing and further processing leads to stationary curves from step 66 in this example.

[34] We show a second numerical example for the LASSO scheme in Figures 7 and 8. Here, for four unknown objects different stages of the iterative process are displayed, which show that the unknown scatterers (again indicated in light grey) are found step by step while the curve contracts.

Figure 7.

We show reconstruction steps 100, 120 and 140 when reconstructing an unknown object which has four separate components.

Figure 8.

We show reconstruction steps 170, 220 and 266 when reconstructing an unknown object which has four separate components.

[35] We need to close with some further remarks. The practical realization of smoothing and splitting has turned out to be important for the performance of the algorithm. The level of smoothing is important to keep the reconstructions of the indicator valid. At the same time smoothing prevents curves from moving inward visualized as in Figure 4. We have implemented the following strategy:

[36] 1. When we do not obtain any further movement of the curve, but still do not have all points at the boundary indicated by the size of the reconstructed probe functional, then we use less smoothing for the further steps. This is controlled by lowering the smoothing parameter (compare equations (24) or (25)). We have multiplied it by 0.5 in this case.

[37] 2. After splitting two domains, we want to smoothen the outcome of the splitting process. We have multiplied the smoothing parameter by 4 and continued with the iterations.

[38] 3. Initially, eigenvalues of the domain Gn caused severe problems to obtain reliable and stable reconstructions of the indicator function. Numerically, we could avoid the eigenvalues by choosing some additional interior curve equation image in Gn which did not allow interior eigenvalues and solving the point source equation on equation image ∪ ∂Gn. An alternative is to switch to the interior impedance problem to obtain a solution to the point source equation (9), an approach suggested by A. Kirsch (Mathematical Research Institute Oberwolfach, private communication, 2011).

[39] The result of this strategy can be seen in the second example, compare in particular Figure 7, where the result of smoothing can be observed.

5. Conclusion

[40] We have surveyed several approaches to implementing probe methods for the reconstruction of objects from the far-field pattern of acoustic or electromagnetic waves. In particular, we have discussed the advantages and disadvantages of the needle approach and domain sampling. With the LASSO scheme we suggested a novel approach which remedies some of the difficulties of former schemes. We have discussed the convergence of the method and shown numerical examples which show the feasibility of the approach.

Acknowledgments

[41] The research has been supported by EPSRC under grants EP/E032419/1 and EP/F033036/1 and by a Leverhulme research fellowship on Electromagnetic Inverse Scattering. The author is grateful to Gen Nakamura, Hokkaido University, Japan, for many helpful discussions and his hospitality during several research visits to Japan. Also, Andreas Kirsch, Karlsruhe Institute of Technology, provided valuable suggestions. I would like to thank the German Meteorological Service (Deutscher Wetterdienst DWD) for supporting this basic research agenda.